The Sound of Chaos

Chaos theory and its little cousin – strange attractors – have been around for a long time. Pictures of chaotic systems and strange attractors abound, and they are a mainstay for computer math experimentalists, although still in the minor leagues relative to the Mandelbrot set.

Most implementations tend to ignore the fact that these systems represent dynamics, that they move and evolve. Still pictures can hide the fact that, for example there are sink states, and that was supposed to represent 20,000 iterations only shows 5,000 because the system hit a fixed point at iteration 5,001.

Years ago I wrote a Windows program to show the dynamics of strange attractors, but that experiment quickly moved over to computing the dynamics of iterated function systems, iterated rational polynomials, and Kleinian groups. And I always wanted to use these systems to generate sounds.

All that remained on the back burner until this year. I recently discovered that the Flash Player now supports on-the-fly sound generation. And an implementation in Flash would conform to my self-imposed mandate to write only software that was browser based. So I downloaded the latest Flex Builder beta from Adobe and set about learning enough Actionscript to get this project going.

It’s been a tough slog because Flex 4 has many departures from Flex 3, in particular the Spark component set, which is sort of the same as Halo, but also different. So hard times for a newb.

Anyways, I’ve got a bare example experiment going, which is in my beta area. It comes with a video that should explain what’s going on.

Math with a soundtrack

There’s something about music or a soundtrack that really enhances what would otherwise be just a silent movie. So I plan to add music to my visualizations as time and inspiration allow. Here’s an example that I did a week or two back. The original movie was 24 seconds long, but now it’s been slowed down to accommodate an atmospheric soundtrack.

Hyperbolic Kaleidoscope from Peter Liepa on Vimeo.

Classroom treat

I just got word that a UK professor was going to show these movies to her hyperbolic geometry class today, as a treat for the last class of the term.

Reminds me of when I was in public school and we got ice cream on the last day of school.

In another day and age, they would have been asked to get out their rulers and compasses and draw the thing, as shown here. Up until recently, that’s representative of the best depiction mathematicians had.

The movie that corresponds to it is here.


I’ve set myself the task of creating a movie that shows a hyperbolic tessellation. This project has been brewing in one form or another for years, but I’ve finally got the time and motivation to at least get started on it. Along the way, we’ll follow background, progress and various tangents. Although the math part is important, the software part will play a role, because at the moment I’ve got too many choices.